The cardinality of the set of discontinuous solutions of a class of functional equations (Q1821279)

The paper deals with the following equation \[ (1)\quad f([f(x)]^ ky+[f(y)]^{\ell}x)=\lambda f(x)f(y) \] where k,\(\ell \in {\mathbb{N}}\) and \(\lambda\in {\mathbb{R}}\) are given and f:\({\mathbb{R}}\to {\mathbb{R}}\) is unknown. The following theorem is proved: equation (1) has \(2^{\aleph}\) discontinuous solutions if a) \(\lambda\geq 0\), b) \(\lambda <0\) and k,\(\ell \in 2{\mathbb{N}}\), c) \(0>\lambda\) is transcendental, d) \(0>\lambda\) is algebraic over \({\mathbb{Q}}\) and its minimal polynomial has a positive root. The author uses in the proof some facts concerning derivation known e.g. form \textit{J. Aczél's} book [Lectures on functional equations and their applications (1966; Zbl 0139.093)].